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Kirill Zainoulline

Publications and source records attributed to Kirill Zainoulline.

At least 37 records · Page 2Linked to original sources
arXiv2016

A Schubert basis in equivariant elliptic cohomology

We address the problem of defining Schubert classes independently of a reduced word in equivariant elliptic cohomology, based on the Kazhdan-Lusztig basis of a corresponding Hecke algebra. We study some basic properties of these classes, and make two important conjectures about them: a positivity conjecture, and the agreement with the topologically defined Schubert classes in the smooth case. We prove some special cases of these conjectures.

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arXiv2015

Towards generalized cohomology Schubert calculus via formal root polynomials

An important combinatorial result in equivariant cohomology and $K$-theory Schubert calculus is represented by the formulas of Billey and Graham-Willems for the localization of Schubert classes at torus fixed points. These formulas work uniformly in all Lie types, and are based on the concept of a root polynomial. In this paper we define formal root polynomials associated with an arbitrary formal group law (and thus a generalized cohomology theory). We focus on the case of the hyperbolic formal group law (corresponding to elliptic cohomology). We study some of the properties of formal root polynomials. We give applications to the efficient computation of the transition matrix between two natural bases of the formal Demazure algebra in the hyperbolic case. As a corollary, we rederive in a transparent and uniform manner the formulas of Billey and Graham-Willems. We also prove the corresponding formula in connective $K$-theory, which seems new, and a duality result in this case. Other applications, including some related to the computation of Bott-Samelson classes in elliptic cohomology, are also discussed.

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arXiv2014

Invariants of degree 3 and torsion in the Chow group of a versal flag

We prove that the group of normalized cohomological invariants of degree 3 modulo the subgroup of semidecomposable invariants of a semisimple split linear algebraic group G is isomorphic to the torsion part of the Chow group of codimension 2 cycles of the respective versal G-flag. In particular, if G is simple, we show that this factor group is isomorphic to the group of indecomposable invariants of G. As an application, we construct nontrivial cohomological classes for indecomposable central simple algebras.

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arXiv2014

Push-pull operators on the formal affine Demazure algebra and its dual

In the present paper we introduce and study the push pull operators on the formal affine Demazure algebra and its dual. As an application we provide a non-degenerate pairing on the dual of the formal affine Demazure algebra which serves as an algebraic counterpart of the natural pairing on the T-equivariant oriented cohomology of G/B induced by multiplication and push-forward to a point. This paper can be viewed as the next step towards the `algebraization program' for equivariant oriented cohomology theories started in arXiv:0905.1341 and continued in arXiv:1208.4114 and arXiv:1209.1676; the general idea being to match cohomology rings of algebraic varieties and elements of classical interest in them (such as classes of Schubert varieties) with algebraic and combinatorial objects that can be introduced in the spirit of [Demazure, Invariants symétriques entiers des groupes de Weyl et torsion, Invent. Math. 21:287-301, 1973] and [Kostant, Kumar, The nil Hecke ring and cohomology of G/P for a Kac-Moody group G, Advances in Math. 62:187-237, 1986].

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arXiv2014

A coproduct structure on the formal affine Demazure algebra

In the present paper we generalize the coproduct structure on nil Hecke rings introduced and studied by Kostant-Kumar to the context of an arbitrary algebraic oriented cohomology theory and its associated formal group law. We then construct an algebraic model of the T-equivariant oriented cohomology of the variety of complete flags.

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arXiv2014

Equivariant oriented cohomology of flag varieties

Given an equivariant oriented cohomology theory $h$, a split reductive group $G$, a maximal torus $T$ in $G$, and a parabolic subgroup $P$ containing $T$, we explain how the $T$-equivariant oriented cohomology ring $h_T(G/P)$ can be identified with the dual of a coalgebra defined using exclusively the root datum of $(G,T)$, a set of simple roots defining $P$ and the formal group law of $h$. In two papers [Push-pull operators on the formal affine Demazure algebra and its dual, arXiv:1312.0019] and [A coproduct structure on the formal affine Demazure algebra, arXiv:1209.1676], we studied the properties of this dual and of some related operators by algebraic and combinatorial methods, without any reference to geometry. The present paper can be viewed as a companion paper, that justifies all the definitions of the algebraic objects and operators by explaining how to match them to equivariant oriented cohomology rings endowed with operators constructed using push-forwards and pull-backs along geometric morphisms. Our main tool is the pull-back to the $T$-fixed points of $G/P$ which injects the cohomology ring in question into a direct product of a finite number of copies of the $T$-equivariant oriented cohomology of a point.

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arXiv2014

On formal Schubert polynomials

Present notes can be viewed as an attempt to extend the notion of Schubert/Grothendieck polynomial to the context of an arbitrary algebraic oriented cohomology theory and, hence, of a commutative one-dimensional formal group law.

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arXiv2014

Formal Hecke algebras and algebraic oriented cohomology theories

In the present paper we generalize the construction of the nil Hecke ring of Kostant-Kumar to the context of an arbitrary algebraic oriented cohomology theory of Levine-Morel and Panin-Smirnov, e.g. to Chow groups, Grothendieck's K_0, connective K-theory, elliptic cohomology, and algebraic cobordism. The resulting object, which we call a formal (affine) Demazure algebra, is parameterized by a one-dimensional commutative formal group law and has the following important property: specialization to the additive and multiplicative periodic formal group laws yields completions of the nil Hecke and the 0-Hecke rings respectively. We also introduce a deformed version of the formal (affine) Demazure algebra, which we call a formal (affine) Hecke algebra. We show that the specialization of the formal (affine) Hecke algebra to the additive and multiplicative periodic formal group laws gives completions of the degenerate (affine) Hecke algebra and the usual (affine) Hecke algebra respectively. We show that all formal affine Demazure algebras (and all formal affine Hecke algebras) become isomorphic over certain coefficient rings, proving an analogue of a result of Lusztig.

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arXiv2013

Invariants, exponents and formal group laws

Let W be the Weyl group of a crystallographic root system acting on the associated weight lattice by reflections. In the present notes we extend the notion of an exponent of the W-action introduced in [Baek-Neher-Zainoulline, arXiv:1106.4332] to the context of an arbitrary algebraic oriented cohomology theory of Levine-Morel, Panin-Smirnov and the associated formal group law. From this point of view the classical Dynkin index of the associated Lie algebra will be the second exponent of the deformation map from the multiplicative to the additive formal group law. We apply this generalized exponent to study the torsion part of an arbitrary oriented cohomology theory of a twisted flag variety.

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arXiv2013

Erratum: On the torsion of Chow groups of twisted Spin-flags

In the erratum we correct a mistake (due to a wrong choice of basic polynomial invariants over Z[1/2]) in the original paper (v1). Using the correct basic polynomial invariants we improve our results and bounds on the annihilator. We also simplify some of the proofs.

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arXiv2012

Exceptional collections of line bundles on projective homogeneous varieties

We construct new examples of exceptional collections of line bundles on the variety of Borel subgroups of a split semisimple linear algebraic group G of rank 2 over a field. We exhibit exceptional collections of the expected length for types A_2 and B_2=C_2 and prove that no such collection exists for type G_2. This settles the question of the existence of full exceptional collections of line bundles on projective homogeneous G-varieties for split linear algebraic groups G of rank at most 2.

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arXiv2011

The J-invariant, Tits algebras and Triality

In the present paper we set up a connection between the indices of the Tits algebras of a simple linear algebraic group $G$ and the degree one parameters of its motivic $J$-invariant. Our main technical tool are the second Chern class map and Grothendieck's $γ$-filtration. As an application we recover some known results on the $J$-invariant of quadratic forms of small dimension; we describe all possible values of the $J$-invariant of an algebra with orthogonal involution up to degree 8 and give explicit examples; we establish several relations between the $J$-invariant of an algebra $A$ with orthogonal involution and the $J$-invariant of the corresponding quadratic form over the function field of the Severi-Brauer variety of $A$.

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arXiv2011

Basic polynomial invariants, fundamental representations and the Chern class map

Consider a crystallographic root system together with its Weyl group $W$ acting on the weight lattice $M$. Let $Z[M]^W$ and $S^*(M)^W$ be the $W$-invariant subrings of the integral group ring $Z[M]$ and the symmetric algebra $S^*(M)$ respectively. A celebrated theorem of Chevalley says that $Z[M]^W$ is a polynomial ring over $Z$ in classes of fundamental representations $w_1,...,w_n$ and $S^*(M)^{W}$ over rational numbers is a polynomial ring in basic polynomial invariants $q_1,...,q_n$, where $n$ is the rank. In the present paper we establish and investigate the relationship between $w_i$'s and $q_i$'s over the integers. As an application we provide an annihilator of the torsion part of the 3rd and the 4th quotients of the Grothendieck gamma-filtration on the variety of Borel subgroups of the associated linear algebraic group.

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arXiv2011

Equivariant pretheories and invariants of torsors

In the present paper we introduce and study the notion of an equivariant pretheory: basic examples include equivariant Chow groups, equivariant K-theory and equivariant algebraic cobordism. To extend this set of examples we define an equivariant (co)homology theory with coefficients in a Rost cycle module and provide a version of Merkurjev's (equivariant K-theory) spectral sequence for such a theory. As an application we generalize the theorem of Karpenko-Merkurjev on G-torsors and rational cycles; to every G-torsor E and a G-equivariant pretheory we associate a graded ring which serves as an invariant of E. In the case of Chow groups this ring encodes the information concerning the motivic J-invariant of E and in the case of Grothendieck's K_0 -- indexes of the respective Tits algebras.

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arXiv2011

Twisted gamma-filtration of a linear algebraic group

In the present notes we introduce and study the twisted gamma-filtration on K_0(G), where G is a split simple linear algebraic group over a field of characteristic prime to the order of the center of G. We apply this filtration to construct torsion elements in the gamma-ring of an inner form of the variety of Borel subgroups of G.

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arXiv2010

The gamma-filtration and the Rost invariant

Let X be the variety of Borel subgroups of a simple and strongly inner linear algebraic group G over a field k. We prove that the torsion part of the second quotient of Grothendieck's gamma-filtration on X is a cyclic group of order the Dynkin index of G. As a byproduct of the proof we obtain an explicit cycle that generates this cyclic group; we provide an upper bound for the torsion of the Chow group of codimension-3 cycles on X; we relate the generating cycle with the Rost invariant and the torsion of the respective generalized Rost motives; we use this cycle to obtain a uniform lower bound for the essential dimension of (almost) all simple linear algebraic groups.

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arXiv2010

Invariants, torsion indices and oriented cohomology of complete flags

In the present notes we generalize the classical work of Demazure [Invariants symétriques entiers des groupes de Weyl et torsion] to arbitrary oriented cohomology theories and formal group laws. Let G be a split semisemiple linear algebraic group over a field and let T be its split maximal torus. We construct a generalized characteristic map relating the so called formal group ring of the character group of T with the cohomology of the variety of Borel subgroups of G. The main result of the paper says that the kernel of this map is generated by W-invariant elements, where W is the Weyl group of G. As one of the applications we provide an algorithm (realized as a Macaulau2 package) which can be used to compute the ring structure of an oriented cohomology (algebraic cobordism, Morava $K$-theories, connective K-theory, Chow groups, K_0, etc.) of a complete flag variety.

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